Differentiable function

A function f(x) is differentiable at the point a if and only if, as x approaches a (which it is never allowed to reach), the value of the quotient:

$\frac{f(x) - f(a)}{(x - a)}$

approaches a limiting value that we call the derivative of the function f(x) at x=a.

There is also the more rigorous $\epsilon-\delta$ definition: a function f is said to be differntiable at point a if ∀$\epsilon>0$ ∃​$\delta>0$ such that if

$|x - a| < \delta\,$

then

$|\frac{f(x) - f(a)}{x-a} - f'(a) | < \epsilon \,$.